Driven diffusion in nanoscaled materials Abstract

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application

Driven diffusion in nanoscaled materials Tony Albers1, Michael Bauer1, Christian von Borczyskowski1*, Frank Gerlach1,3, Mario Heidernätsch1, Jörg Kärger2, Daria Kondrashova2, Günter Radons1#, Sebastian Schubert1,4, Alexander Shakhov2, Daniela Täuber1,5, Rustem Valiullin2§, Philipp Zeigermann2 1Institute 2Institute

of Physics, Technische Universität Chemnitz, Chemnitz, Germany for Experimental Physics I, Universität Leipzig, Leipzig, Germany 3now

4now

at Fibotec Fiberoptics GmbH, Meiningen, Germany

at MPI for Biophysical Chemistry, Göttingen, Germany 5now

at Lund University, Lund, Sweden

*[email protected], #[email protected], §[email protected]

Abstract Mass transfer processes in which specific interactions with environments lead to complex diffusion patterns, such as the occurrence of transient sub-diffusive behaviors or of heterogeneous diffusion, were studied by means of two different experimental techniques, namely single-particle tracking operating with single molecules and nuclear magnetic resonance operating with large molecular ensembles. As an important point, the combined application of these techniques allowed for a deeper insight into the microscopic diffusion mechanism in such complex systems, including those with broken ergodicity. Particle tracking concentrated on the “Influence of substrate surface properties on heterogeneous diffusion of probe molecules in ultrathin liquid films”. The mobility of liquids at solid-liquid interfaces is influenced by substrate heterogeneities. Here we study the distribution of surface silanols on differently treated silicon wafers with thermal oxide by confocal florescence microscopy of adsorbed Rhodamine G molecules. We further investigate the influence of the substrate properties on probe molecule diffusion in ultrathin liquid TEHOS films by single molecule tracking. The results are compared to simulations of two-layer diffusion employing heterogeneous substrates. Nuclear magnetic resonance has been applied to study translational diffusion of small organic molecules in nanopores and of polymer globules in the presence of larger polymer species. In both cases, the experiments revealed the occurrence of normal diffusion on the time scale of NMR experiments from ten to hundreds of milliseconds. While single particle tracking revealed the identical diffusivities for the former case, thus experimentally confirming the validity of the ergodicity theorem for diffusion, the discrepancies were noted for the latter case. More complex behavior revealing non-ergodic

© 2015, T. Albers et al. diffusion-fundamentals.org 23 (2015) 3, pp 1–25

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behavior for propagation of solid-liquid interfaces in disordered nanopores has further been studied using nuclear magnetic resonance cryoporometry. A common basis for comparing and analyzing the experimental observables accessed by the two methods is the distribution of diffusivities, which provides the probability of observing a given diffusivity fluctuation along a trajectory or in an ensemble. An overview of its properties is given and the advantages in analyzing heterogeneous, anisotropic, or anomalous diffusion processes are elaborated. Keywords: Surface silanols, ultrathin liquid films, single molecule detection, tracer diffusion, NMR, solid-liquid interface, simulation, distribution of diffusivities, heterogeneous diffusion, anisotropy, ergodicity

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Influence of substrate surface properties on heterogeneous diffusion of probe molecules in ultrathin liquid films

1.1 Introduction The mobility of liquids at solid-liquid interfaces is influenced by substrate heterogeneities. Here we study the distribution of surface silanols on differently treated silicon wafers with thermal oxide by confocal fluorescence microscopy of adsorbed Rhodamine G molecules. Silicon wafers with native or thermal oxide are widely used in technical applications, for example in solar cells or for chip fabrication [1]. The performance of such devices is influenced by surface chemistry [2] and by local interface properties [3]. In particular, silica surfaces are covered by hydroxyl groups (silanols) and water, depending on fabrication and temperature treatment [4, 5]. Due to the high fabrication temperature (1000 °C) of the thermal oxide, the silanol coverage is significantly lower compared to fumed silica [5]. Yet, the amount and distribution of surface silanols is known to affect nanotechnical fabrication routines, for example nanolithography [6] and coatings with self-assembled monolayers [7]. However, up to now only average silanol coverages were known from ensemble measurements on macroscopic scales [4]. Here we use laser scanning confocal microscopy to investigate the distribution of fluorescent rhodamine 6G (R6G) adsorbed on 100 nm thermal oxide of Si wafers. According to Iler [8] charged dye molecules preferentially physisorb to surface silanols. Thus, the derived spatial distribution of adsorbed R6G molecules unravels the distribution of surface silanols on a submicron range. Interactions at solid-liquid interfaces influence dynamic properties of fluids [9, 10] and guest molecules therein, which in particular plays a role in surface processes [11]. Optical investigation of single molecule diffusion in ultrathin liquid films on quartz cover slips revealed anisotropic diffusion of probe molecules [12]. This has been related to liquid layering at solid-liquid interfaces [13, 14]. On the other hand, also heterogeneous interface chemistry and morphology causes anisotropies in probe molecule diffusion, see for example Figure 1 [15], showing trajectories with position related heterogeneities obtained from diffusing Rhodamine B (RhB) molecules in ultrathin TEHOS films on silicon substrates with 100 nm thermally grown SiO2.


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Figure 1: (left) Spatial plot of several immobile (bright areas) and three mobile (colored trajectories) obtained for a 10 nm thick TEHOS film on SiO2 doped with RhB. (right) Diffusivities 𝑑diff over total time 𝑡 for light green and purple trajectory from the left image. The center of the solid and dashed circles in the left image mark areas with repeated periods of immobilization. The corresponding sets of periods are marked by solid and dashed arrows in the right image [15].

We use a homebuilt wide-field microscope [15] to study the influence of surface silanols on tracer diffusion in ultrathin liquid films. A common way to analyze diffusion experiments is to calculate mean square displacements (MSD) along detected trajectories [16]. However, long trajectories are necessary for good statistics [17]. In particular, heterogeneous or anomalous diffusion may be concealed due to the strong time averaging [18]. Moreover, the length of observed trajectories of single dye molecules is often limited by photobleaching and fluorescence intermittency [19]. Improved statistics can be achieved by detecting square displacements (Δ𝑟)2 for succeeding time steps at a fixed time lag 𝜏 [20]. Thus, probability distributions 𝐶(𝐷)𝜏 of time scaled (Δ𝑟)2 between succeeding steps (diffusivities 𝑑diff =

(Δ𝑟)2 /(4𝜏)) provide a promising alternative [15, 21], since time averaging is realized and statistics are increased. 𝐶(𝐷)𝜏 is defined as the probability of finding a diffusivity 𝑑diff > 𝐷. By scaling the squared displacements by the corresponding time lag 𝜏 the trivial dependence on 𝜏 is removed and thus facilitates the comparability for different 𝜏. However, for heterogeneous diffusion the shape of the probability distribution of diffusivities may still depend on 𝜏 [22]. For this reason, 𝜏 is noted as index. In case of homogeneous diffusion with diffusion coefficient 𝐷, the cumulative probability distribution of diffusivities amounts to

𝐶(𝐷)𝜏 = exp⁡(−𝐷/𝐷0 ).

(1)

In this case, probability distributions of diffusivities show a linear behavior in semi-log plots. In case of heterogeneous or anomalous diffusion, deviations from the mono-exponential function will appear. The further analysis then depends on the origin of such deviations [22]. Our findings show that the most regular distribution of surface silanols is obtained after etching the substrates in H2O2:H2SO4 (piranha) solution. Larger silanol clusters (on hydroxylated substrates) and larger distances between clusters (on tempered substrates) both enhance the heterogeneity of probe molecule diffusion.

1.2 Experimental Si(100) wafers with 100 nm thermal oxide (dry O2/HCl processed at 1000 °C) were obtained from the Center for Microtechnologies (ZfM, Chemnitz, Germany). In particular, for optical single molecule


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investigations clean substrates are needed. In addition to commonly applied cleaning procedures [15], some substrates were boiled in ultraclean water for 20 h (“hydroxylated”), while others were tempered for 0.5 h at 800 °C in air (“tempered”). Substrates which underwent no additional treatment besides cleaning were used for comparison (“piranha”).

1.2.1 Determination of local silanol densities The substrates were dipcoated in solutions of 10-6 mol/l Rhodamine 6G (R6G, Radiant Dyes) in ethanol using a stand-alone dipcoater (KSV-DX2, KSV Instruments). Similar concentrated solutions of R6G in toluene and in n-hexane were used for comparison (all solvents of spectroscopic grade, Merck). The withdrawal speed was 5 mm/min. Thereby a thin film of the solution is generated on the substrate. This film evaporates within 30 minutes, while dye molecules remain on the substrate at silanol binding sites. Fluorescence images of the R6G distribution were recorded with a homebuilt laser scanning confocal microscope [23]. Confocal images (100⁡±⁡20 µm2) were analyzed using WSxM 5.0 [24]. First, an intensity threshold was set for each image at its mean intensity minus one standard deviation. Then the images were smoothed with a Gaussian profile. From the processed images cluster sizes and coverages were determined using a cluster threshold size of 0.05 µm2 [15, 25].

1.2.2 Probe diffusion measurements Probe molecule diffusion at the solid liquid interface was investigated using ultrathin liquid films of Tetrakis-2-ethylhexoxy-silane (TEHOS, abcr). Due to the small film thickness of 5⁡±⁡1 nm probe molecules remain close to the substrate. TEHOS films were obtained by dipcoating the substrates in solutions of few ppt TEHOS in n-hexane (spectroscopic grade, Merck) [15]. The solutions were doped with fluorescent Rhodamine B (RhB) resulting in a nanomolar concentration of dye molecules in the TEHOS film. The diffusion of RhB was recorded with a wide-field microscope with epi-illumination [15]. Since the z-focus (about 1 µm) is considerably larger than the film thickness, the 3D motion is projected onto a 2D motion parallel to the substrate. Thus, the obtained diffusion coefficients are effective diffusion coefficients parallel to the substrate. Series of 1800 succeeding images were recorded with a CCD camera (iXon DU 885K, Andor) at a frame rate of 1 fps, resulting in 30 min observation time for each measurement. The diffusion tracks were analyzed using home-written C/C++ programs [15, 26]. Immobile RhB molecules were excluded from further analysis.

1.3 Results and Discussion 1.3.1 Variation of Hydroxylation Fluorescence intensities from R6G after evaporation of the solvent show an inhomogeneous distribution. In Figure 2 fluorescence intensities from three differently treated substrates are shown after evaporation of ethanol. Raw images have been processed with WSxM 5.0 [24]. Similar results were obtained using toluene and n-hexane as solvents (not shown). Thus, the distribution of R6G molecules on the thermal oxide is not influenced by the kind of solvent, but is determined by the properties of the substrate surface.


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Figure 2: Clustered distribution of R6G fluorescence intensities on thermal SiO2 after (left) tempering at 800 °C in air, (middle) cleaning in piranha with no further treatment, and (right) hydroxylation in water.

The results from image processing are given in Table 1. As can be seen, there is only a slight modification of the silanol coverage from 46.4 % for the tempered substrate to 53.5 % for the hydroxylated substrate. The largest average cluster size (1.97⁡±⁡0.5 µm2) is observed for the hydroxylated substrate. The thermal oxide tempered for 30 min at 800 °C shows only a slightly smaller average cluster size (1.42⁡±⁡0.3 µm2), while the smallest average cluster size (0.64⁡±⁡0.2 µm2), is obtained for the substrate which was etched in piranha followed by no further treatment. Hydroxylation increases the size of the silanol clusters, while the nearest neighbor distance between clusters remains at a similar value (0.66⁡±⁡0.3 µm) as for the substrate after piranha treatment (0.64⁡±⁡0.2 µm). Tempering the thermal oxide at high temperatures reduces primarily the amount of smaller clusters, thus leading to a threefold nearest neighbor distance (1.75⁡±⁡0.5 µm) as compared to the other substrate treatments. This finding is in agreement with the assumption that surface silanols are preferentially generated and diminished at the edge of existing silanol clusters [4, 8]. The most regular distribution of surface silanols on thermal SiO2 is achieved by etching in piranha, because the etching not only influences the density of surface silanols, but also affects the underlying SiO 2.

Substrate treatment

tempered

piranha

hydroxylated

Investigated area [µm2] # of clusters in area Coverage [%] Average cluster size and standard deviation [µm2] Nearest neighbour distance [µm]

110 36 46.4 1.42⁡±⁡0.3 1.75⁡±⁡0.5

110 85 49.4 0.64⁡±⁡0.2 0.64⁡±⁡0.2

81 22 53.5 1.97⁡±⁡0.5 0.66⁡±⁡0.3

Table 1: Local silanol distribution on differently treated thermal SiO2.

1.3.2 Probe diffusion in ultrathin TEHOS films The varying distribution of surface silanols is not only identified in the above described (decoration) experiments, but has also an influence on probe molecule diffusion in ultrathin liquid films. Figure 3 (bottom, experimental data) shows 𝐶(𝐷)𝜏 obtained for diffusing RhB in ultrathin TEHOS films on thermal oxide for various substrate treatments. Immobile molecules have been omitted according to the position accuracy depending on the particular signal to noise ratio.


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C(D)=1s

D [m2/s]

D [m2/s]

D [m2/s]

Figure 3: (bottom) Cumulative probability distributions of diffusivities 𝐶(𝐷)𝜏 from simulations (●) and experiment (○) together with fits according to (2), for simulations (dotted line) and experiment (dashed line) obtained for different experimentally determined silanol distributions such as shown in Figure 2: (left) tempered, (middle) piranha treated and (right) hydroxylated substrates. (top) heterogeneous substrate coverage used for the simulation, areas with slow diffusion are black. (middle) time integrated fluorescence intensities obtained from tracking analysis with tracking.sh.

As can be seen, the 𝐶(𝐷)𝜏 (Figure 3 bottom) deviate from a mono-exponential behavior for all three kinds of substrate treatments. The analysis of heterogeneous diffusion is not straightforward and the diffusion coefficients may be derived from fitting experimental data only in some particular cases using only two distinct components [15, 22, 26]. Nevertheless, an approximation using a sum of two exponentials, Eq. (2), provides further insights into probe diffusion behavior.

𝐶(𝐷)𝜏 = 𝐴1 exp(−𝐷/𝐷1 ) + 𝐴2 exp⁡(−𝐷/𝐷2 ), with 𝐴1 + 𝐴2 = 1.

(2)

This simplified approach has been used since we suggest a simple 2-layer model as described in Figures 4 and 5. Naturally, more complex fitting functions would fit the data more closely. The results from fitting the data with (2) are given in Table 2. In case of the piranha treated substrate, the biexponential approximation fits the experimental data quite well, whereas for the hydroxylated substrates larger diffusivities are observed. These deviations are even more pronounced in case of the tempered substrate. However, deviations only occur in the amplitude range of 10 -2 to 10-3. Moreover, only a few


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data points can be collected for fast diffusion in case of high silanol coverage (middle and right graph). The amplitudes 𝐴1 for the slow component increase from the “tempered” to the “hydroxylated” substrate thereby following the increase of coverage with surface silanols as given in Table 1. These findings can be interpreted using a simple lateral two-region model as depicted in Figure 4. We assume that probe molecules diffuse on top of the silanol clusters with an effective diffusion coefficient 𝐷slow, whereas they diffuse with 𝐷fast between the clusters. The slow diffusion on the silanol clusters is caused by adsorption/desorption kinetics during the image exposure time. In case of probe molecules with two possible H-bonding sites (as it is the case for RhB), also some kind of surface gliding by successive building and breaking of H-bonds with neighboring surface silanols may be considered, as suggested by Honciuc et al. [27]. The here described experiment did not allow for discrimination between fast re-adsorbtion dynamics and surface gliding. The slow effective diffusion will depend on the particular probe molecule chemistry.

𝐷fast cannot be directly derived from the silanol density distribution. The diffusion coefficient of RhB in bulk TEHOS is about 50 µm2/s [15]. Thus, the probe molecule would diffuse across an effective area of about 50 µm2 during an exposure time of 1 s which is considerably larger than the average area separating neighboring silanol clusters (see Figure 2). Therefore probe molecules diffuse with 𝐷fast only part of the image exposure time. The remaining time they will diffuse with 𝐷slow on top of the silanol clusters.

Figure 4. Influence of interface heterogeneity on probe molecule diffusion including physi- or chemisorption [15].

For this reason, the geometry of the silanol clusters and the interspacing region has an essential influence on the probability distribution of diffusivities. On the piranha treated substrate, silanols constitute many small clusters resulting in a narrow cluster size distribution with a small and quite regular interspacing distance. Thus, the correlated diffusivities show an almost mono-exponential dependence for diffusion coefficient 𝐷2 . In this case, 𝐷2 resembles the average diffusion coefficient for fast probe molecules. On contrary, in case of the tempered and the hydroxylated substrate, silanol cluster sizes and the interspacing distances show a much broader distribution. Therefore, diffusivities are not monoexponentially distributed. For the tempered substrate, the much larger interspacing areas lead to a stronger contribution from longer transients, increasing the probability for larger diffusivities.


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1.3.3 Simulation of diffusion influenced by surface heterogeneities To further evaluate the influence of substrate surface heterogeneities on diffusion coefficients, we performed simulations using a vertical two layer model. Within the software package dsaa, Heidernätsch had developed a simulation tool which allowed to incorporate structured substrates into a multi-layer model and simulates also wide-field acquisition of probe molecule diffusion [26]. Figure 5 (top) illustrates two-layer diffusion for a model substrate (top, right) containing stripes with reduced mobility (black). In the model (top, left), a reduced diffusion coefficient 𝐷1a is used in layer 1 on the black areas of the substrate, while on the white areas, the diffusion coefficient 𝐷1b was set equal to the top-layer diffusion coefficient 𝐷2 . Figure 5 (bottom) shows the results from analyzing simulated wide-field data with the tracking package tracking.sh [12, 15]. For this simulation the diffusion coefficients have been set to

𝐷2∗ = 𝐷1b∗ =⁡1 µm2/s and 𝐷1a∗ =⁡0.01 µm2/s with exchange rates 𝑓12 =⁡0.5 s-1 and 𝑓21 =⁡0.01 s-1 between the upper and lower layer and a time step d𝑡 =⁡0.1 ms. The wide-field acquisition was simulated with an exposure time 𝛥𝑡 =⁡1 s for each frame and a total number of 1800 subsequent frames. Figure 5 (bottom right) shows simulated intensities integrated over all 1800 frames on a scale ranging from low (blue) to high (white) fluorescence intensities. The stripes with slow diffusion (black) in layer 1 correspond to high fluorescence intensities, separated by low intensity stripes marking the areas with fast diffusion. The lower intensity between stripes of slow diffusion is related to the fact that for fast diffusing molecules the detected intensity will be smeared out over a wider area during one time step. Additionally, the stationary distribution of probe molecules yields a higher probability for molecules on the stripes with correlated slow diffusion. Figure 5 (bottom, left) shows the simulation of the cumulated probability distribution of diffusivities from all detected trajectories together with a fit according to (2). On the time scale of the observation, the diffusion is heterogeneous, as can be seen from the fit deviating from a straight line. We obtain effective diffusion coefficients 𝐷1 =⁡0.012 µm2/s and 𝐷2 =⁡0.055 µm2/s with amplitudes 𝐴1 =⁡0.9 and

𝐴2 =⁡0.1, respectively. The obtained effective diffusion coefficients are intermediates between the diffusion coefficients 𝐷1a∗ =⁡0.01 µm2/s and 𝐷2∗ = 𝐷1b∗ =⁡1 µm2/s used for the simulation and do not correspond to the real diffusion coefficients. While 𝐷slow = 𝐷1 =⁡0.012 µm2/s gives a reasonable value for diffusion across silanol clusters with short excursion into the upper film region, Dfast cannot be retrieved from these simulations, whereby 𝐷2 =⁡0.055 µm2/s gives rather an average for very fast and the slow diffusion. To proceed further towards a more realistic silanol related cluster model, we used the experimentally obtained silanol distributions (see Figure 2) as heterogeneous substrates as a base for the simulations using the same diffusion parameters as for the previously described model simulations. Figure 3 (top) shows the heterogeneous substrates with (left) tempered, (middle) piranha treated, and (right) hydroxylated surface. Again, areas of slow diffusion are shown in black. The related integrated intensities from analysis of the simulated acquisitions using tracking.sh are shown in Figure 3 (middle). Figure 3 (bottom) shows probability distributions 𝐶(𝐷)𝜏 obtained from the detected trajectories from simulation (●) and in comparison to experimental (○) diffusion of RhB in ultrathin TEHOS films on the differently treated substrates. Fitting data are collected in Table 2.


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C(D)=1s

D Figure 5: Structured model substrate (top, left) two-layer model with a (top, right) substrate with alternating distributions of 𝐷1b (slow) and 𝐷1a (fast) diffusion. (bottom, left) Simulated cumulative distribution of diffusivities with a fit (dashed line) according to (2); (bottom, right) accumulated fluorescence intensity.

𝑪(𝒅𝐝𝐢𝐟𝐟, )𝝉

𝑨𝟏

𝑫𝟏 [µm2/s]

𝑨𝟐

𝑫𝟐 [µm2/s]

tempered, experiment

0.61⁡±⁡0.02

0.012⁡±⁡0.002

0.39⁡±⁡0.02

0.13⁡±⁡0.02

tempered, simulation

0.95⁡±⁡0.02

0.028⁡±⁡0.005

0.05⁡±⁡0.02

0.12⁡±⁡0.02

piranha, experiment

0.68⁡±⁡0.02

0.015⁡±⁡0.003

0.32⁡±⁡0.02

0.09⁡±⁡0.01

piranha, simulation

0.95⁡±⁡0.02

0.028⁡±⁡0.005

0.05⁡±⁡0.02

0.12⁡±⁡0.02

hydroxyl., experiment

0.80⁡±⁡0.02

0.010⁡±⁡0.002

0.20⁡±⁡0.02

0.13⁡±⁡0.02

hydroxyl., simulation

0.95⁡±⁡0.02

0.030⁡±⁡0.005

0.05⁡±⁡0.02

0.15⁡±⁡0.02

Table 2: Parameters from bi-exponential fits according to (2) to the experimental and simulated diffusivities 𝐶(𝐷)𝜏 shown in Figure 3.

The slow diffusion on the silanol clusters appears as areas with high fluorescence intensities (white) in the integrated images (Figure 3 middle). Large areas without clusters appear dark (blue). Fluorescence intensity variations are also found in integrated images from experimentally determined silanol cluster distributions (see Figure 1 left), pointing to an influence from heterogeneous silanol distributions on probe molecule diffusion. A similarity between fluorescence intensity distributions according to probe molecule diffusion (middle) and decorated silanol cluster distribution (top) is clearly observed.

𝐶(𝐷)𝜏 obtained from the simulations show tentatively a larger amplitude of the slow component as compared to the experimental ones, see Figure 3 (bottom) and Table 2. In contrast to simulations, the experimental data contain immobile molecules. In general, fitting diffusivity distributions 𝐶(𝐷)𝜏 yields diffusion coefficients of comparable magnitude both for simulations and experiments for all three © 2015, T. Albers et al. diffusion-fundamentals.org 23 (2015) 3, pp 1–25

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substrate treatments. The best overall agreement between simulation and experiment is obtained for the piranha treated substrate. Here the almost regularly distributed small silanol clusters result in a rather homogeneous substrate surface leading to a more homogeneous lateral diffusion. Consequently, the slow and fast components obtained from fits to experimental data yield the smallest difference with

𝐷1 =⁡0.015 µm2/s and 𝐷2 =⁡0.09 µm2/s (see Table 2). The largest deviation between 𝐶(𝐷)𝜏 from experiment and simulation is found for the tempered substrate (see Figure 3 bottom left). In this case the simulation can be described nearly bi-exponential. According to Table 1 the nearest neighbor distance is in the tempered case much larger than for the other 2 surfaces. However, the simulation is – if restricting to a simple bi-exponentially approximation – not very sensitive to the kind of substrate. Additionally, the fitted amplitudes of the fast component are with respect to the experiment underestimated, especially for the tempered substrate for which an apparently third still faster component does not show up at all. The reason for these deviations is probably due to the fact that the simple approximation of only 2 diffusion coefficients and exchange rates, respectively, does not hold. However, in a first step we intended to keep the simulation model as simple as possible. We suggest that the exchange rates between the layers depend on the underlying structure itself and the ratio 𝑓21 /𝑓12 should be increased for the tempered substrate. Moreover, the diffusion in the upper film layer and the silanol-free areas was set to 𝐷2∗ =⁡1 µm2/s for the simulation. However, the bulk diffusion coefficient of RhB in TEHOS is about 50 µm2/s [15], which is much faster than 1 µm2/s set for diffusion in the upper film layer. Keeping in mind that all diffusion parameters (including exchange rates 𝑓𝑖𝑗 ) have been assumed to be the same for all 3 substrates the general trend of the simulations of 𝐶(𝐷)𝜏 supports the concept of slowed lateral diffusion on surface silanol clusters. In this study we have used surface silanol distributions on differently treated silicon substrates with thermally grown oxide to show how substrate surface heterogeneities influence probe molecule diffusion in ultrathin liquid films. Increasing the heterogeneity by either increasing the average silanol cluster size or the mean distance between the clusters, enhances the heterogeneity in probe molecule diffusion. Simulations employing an oversimplified two-layer model with a (realistic) heterogeneous substrate structure support the concept of slow and fast lateral diffusion across areas on the substrate containing silanol clusters and silanol-free areas, respectively. However, the simulations have to be extended to allow for structure related variations of the simulation parameters. Two further studies not discussed here, investigate the influence of heterogeneous substrate structures on diffusion of nanoscale Ag clusters in surface water layers on silicon oxide substrates [28, 29]. Additionally to the superficial heterogeneities, also long-range interactions, as for example van der Waals forces, have an impact on interfacial mobility and probe molecule diffusion [30]. For intrinsically structured materials such as liquid crystals, there is an additional impact of structure related changes at solid-liquid interfaces on probe molecule diffusion [31–33].


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2

Comparing transport properties obtained using single particle and ensemble observations

2.1 Introduction Mass transfer processes occurring in materials with specific interactions may lead to complex diffusion behavior, such as the occurrence of transient sub-diffusion or of heterogeneous diffusion. Diffusion is the irregular, omnipresent motion of the elementary constituents of matter. Historically, diffusion measurements were based on the observations of large ensembles of diffusing particles [34]. The recently emerged single-particle tracking (SPT) or fluorescence correlation spectroscopy (FCS), in which either single molecules or their small ensembles are traced, have provided a totally new view of diffusion [35–37]. At the same time, these developments posed a problem of inter-relating the messages delivered by these two classes of experimental techniques. The latter was in the focus of the present work. The results and their discussions presented here concern several selected systems, in which the traced markers were small molecules [38, 39], polymer globules [40], and crystal-fluid interfaces in channel-like disordered pores [41–43]. These three different systems considered represented three different situations in which the relevant characteristic time scales of the underlying transport processes were (i) notably shorter, (ii) of the order of, and (iii) notably longer than the experimental time scale, respectively. In this way, with the former (i) and the latter situations (iii) we have mimicked ergodic and non-ergodic systems, respectively; for the intermediate case (ii), according to the literature data anomalous diffusion consistent with the fractional Brownian motion or obstructed diffusion was anticipated to occur [44]. To address these problems experimentally, we used two different techniques of NMR, namely pulsed field gradient (PFG) NMR [45] to directly probe the mean square displacements (MSD) and NMR cryoporometry [46] to follow propagation of the solid-liquid interfaces in nanochannels. The thus assessed quantities represented ones averaged over large ensembles. The results obtained were compared to those obtained either using SPT or FCS. Starting with the most simple system exhibiting normal diffusion we experimentally demonstrate the validity of the ergodic theorem for diffusion. For more complex systems, we report on the discrepancies existing between the two methods and discuss the underlying reasons.

2.2 Results and discussion 2.2.1 Experimental confirmation of the ergodic theorem As a trivial, but as a necessary and experimentally challenging step, before any studies in which timeand ensemble-averaged transport quantities are correlated, one shall be concerned with the experimental demonstration of the validity of the ergodic theorem for systems, in which it is expected to be valid. In these systems, all microscopic time scales should be notably shorter than the experimental one. Despite being generally accepted, the proof of the equivalency of the ensemble- and time-averaged MSD,

〈𝑟 2 (𝑡)〉ensemble = 〈𝑟 2 (𝑡)〉time ,


(3)

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remained not addressed experimentally. Mainly, this was caused by the mutually contradicting measuring conditions inherent in the two experimental approaches, in particular in NMR and in SPT. In SPT, the trajectories of the diffusing single molecules are constructed by fitting the positions of the molecules over time [35]. Therefore the fluorescence signals of the molecules have to be clearly separated from each other, demanding their very low concentrations. Additionally the measurements are limited by the signal-to-noise ratio, which is determined by the brightness of the dye molecules and the integration time. Hence, there is an upper limit for the detectable diffusivity in SPT. Exactly the opposite conditions, namely high concentrations (for generating sufficiently strong NMR signal requiring at least 1018 nuclear spins in the samples) and high diffusivities (for giving rise to observable displacements) must be fulfilled for PFG NMR [45].

Figure 6: (a) Normalized spin-echo diffusion attenuations measured for Atto532 dissolved in deuterated methanol in nanoporous glass using PFG NMR. The two slopes correspond to the Atto532 diffusivities in the nanopores and in the excess bulk phase. (b) Cumulative distributions of the diffusivities measured using SPT (from Ref. [38]).

In order to match the measurements conditions for PFG NMR and SPT studies, we have applied nanoporous glass as a host system for the solution of Atto532 dye molecules in an organic solvent as a guest ensemble. The diffusivity of Atto532 in a nanoporous glass depends on their concentration. For high concentrations it is governed by guest-guest and host-guest interactions, whereas for low concentrations the latter mechanism dominates. While single-molecule experiments were performed in the low-concentration regime (~⁡10-11 mol/l), we managed to reduce the concentration in the PFG NMR experiments to approach this concentration regime corresponding to about 1018 protons of Atto532 in the sample contributing to the NMR signal. Further on, by purposeful tuning the pore diameter of the nanoporous glass down to 3 nm, the diffusivity of Atto532 was adjusted to a range of the values accessible by both techniques. As an example, Figures 6a and 6b show the NMR signal diffusion attenuation and the cumulative diffusivity distribution (defined in Section 3.1) measured for identical samples and under identical conditions using PFG NMR and SPT, respectively. They yield within the experimental accuracy the equal average diffusivities of Atto532 in the nanopores of (1.2⁡±⁡0.5)⁡⋅⁡10-11 m2/s. For the first time, single-molecule and ensemble diffusion measurements were found to experimentally confirm the hypothesis of ergodicity for a simple system exhibiting normal diffusion. With this important step being made, the experimental approach developed can be extended to more


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complex systems, including those for which Eq. (3) may not hold. Our current activities concern, in particular, with systems exhibiting normal diffusion but having broad distributions of diffusivities of the diffusing species. This can be observed, e.g., in diluted polymer solutions with high polydispersity indices. Intrinsic peculiarities inherent in the two experimental methods render the distributions of the measured diffusivities to deviate from each other. Elucidating the particular mechanisms for this observation is currently under progress.

2.2.2 Systems displaying anomalous diffusion With the experiments described in the preceding section, the two so-far separated worlds of diffusion measurements have been brought together for a situation where the rules of normal diffusion are obeyed. However, single particle observations of, e.g., biological systems [47, 48] often seem to exhibit patterns of anomalous diffusion in which

〈𝑟 2 (𝑡)〉 ∝ 𝑡 𝛼 ,

(4)

where  𝐷 II which is typical for elongated molecules. The difference between these cases is depicted in Figure 9.

Figure 9: Distributions of diffusivities of homogeneous anisotropic diffusion processes in three dimensions. For uniaxial molecules the distribution is given by Eq. (14), where the curvature after the peak changes from concave for disc-shaped molecules (blue squares; 𝐷 I = 1, 𝐷 II = 5) to convex in the rod-like shapes (violet pentagons; 𝐷 I = 5, 𝐷II = 1). A further general anisotropic process (green circles; 𝐷1 = 5, 𝐷2 = 3, 𝐷3 = 1) obviously shows a qualitative deviation from the isotropic process with the same asymptotic decay (black triangles; 𝐷𝑐 = 5) which is always concave. The inset depicts the identical asymptotic decays of all processes.

The moments 𝑀1 and 𝑀2 of Eq. (14) are obtained from Eq. (8). These equations can be solved to 1

̌ by 𝐷 I = 𝑀1 ∓ √3𝑀2 − 5𝑀12 and 𝐷 II = 𝑀1 ± √3𝑀2 − 5𝑀12, where the sign obtain the eigenvalues of 𝐃 2

5

has to be chosen to satisfy the constraint of positive diffusion coefficients. For 𝑀12 < 𝑀2 < 2𝑀12 both 3

variants of the signs result in positive diffusion coefficients. Then the third moment 𝑀3 ⁡has to be calculated from each pair of 𝐷 I and 𝐷 II and coincides with the values determined from the data if the correct pair is chosen. Hence, different diffusion coefficients can yield distributions of diffusivities with


19

5

identical first and second moments respectively. For vanishing anisotropy, 𝑀2 → 𝑀12 and 𝐷 I and 𝐷 II 3

approach each other. Hence the single isotropic diffusion coefficient is directly given by the first moment. Close to that limit a decision of the correct pair by 𝑀3 cannot be obtained since both values do not differ significantly.

3.2.2 Processes with extended state space Evaluating many particles simultaneously can be interpreted as the analysis of a process in an extended state space. As an example, current research addresses the tracking of the motion of elongated molecules, such as deoxyribonucleic acid (DNA), which is labeled with fluorescent dyes at both ends [62]. By combining the positions of both dyes into one state vector a process with extended state space is observed effectively. In this example the contributing positions are highly correlated and do not diffuse independently. The resulting process from simultaneous observation comprises 𝑑 -dimensional trajectories of 𝑁 particles. The corresponding state vector 𝐱(𝑡) of the extended state space with effective dimensionality 𝑑eff = 𝑁𝑑 of such a process can be interpreted as an 𝑑eff -dimensional trajectory. It should be noted that the considerations are independent of the actual structure of the extended state space, i.e., even processes of different dimensionality can contribute. The diffusivities of such processes are determined according to Eq. (7) where the dimensionality 𝑑 is substituted by the effective one, 𝑑eff . In the simplest case, many particles diffuse simultaneously and independently. Trajectories in the extended state space of non-interacting particles are closely related to anisotropic processes. In general, each of the 𝑁 independent trajectories is characterized by a different diffusion coefficient, because they may belong to different species. This leads to different diffusive properties in the various directions of the state space. Hence, the distribution of diffusivities can be obtained analogously to anisotropic processes [61], e.g., from the convolution of the contributing processes by 𝑝(𝐷, 𝜏) =

{𝑝1 ∗ ⋯ ∗ 𝑝𝑁 }(𝐷, 𝜏). As an example, consider the simultaneous tracking of two non-interacting twodimensional isotropic processes with different diffusion constants 𝐷1 and 𝐷2 , respectively, which may stem from the particles diffusing in regions with distinct transport properties, or, from different conformations of a molecule. Such a process is characterized by a diffusion tensor with two two-fold degenerate eigenvalues given by 𝐷1 and 𝐷2 . The convolution of the two contributing distributions of diffusivities, which are given by Eq. (9) with 𝑑 = 2 and 𝐷𝑐 = 𝐷1 ⁄2 or 𝐷𝑐 = 𝐷2 ⁄2, respectively, yields

2𝐷 2𝐷 exp (− 𝐷 ) − exp (− 𝐷 ) 1 2 𝑝(𝐷) = 2 . 𝐷1 − 𝐷2

(15)

Since the asymptotic exponential decay defined in Eq. (10), is determined by the larger diffusion coefficient, 𝐷∞ = max(𝐷1 , 𝐷2 ) > 〈𝐷〉 = (𝐷1 + 𝐷2 )/2 holds. Hence, the anisotropy measure Eq. (11) does not vanish and clearly indicates an anisotropic process. Of course, in the limit 𝐷1 → 𝐷2 the 4dimensional isotropic process with the distribution of diffusivities given by the 𝜒42 -distribution of Eq. (9) is recovered.


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3.2.3 Heterogeneous processes The examples we have presented so far, resulted in a distribution of diffusivities 𝑝(𝐷, 𝜏), which turned out to be independent of ⁡𝜏, i.e. 𝑝(𝐷, 𝜏) = 𝑝(𝐷). This is a consequence of the homogeneity and the presence of only one simple scaling law in the previous examples. But for instance, in the example of diffusion of a pair of particles, it is easy to show that the presence of e.g. harmonic interactions already leads to a non-trivial 𝜏-dependence of 𝑝(𝐷, 𝜏), which vanishes only in the limit 𝜏 → ∞. An analytically solvable example of such a non-trivial 𝜏-dependence of 𝑝(𝐷, 𝜏) was worked out for a two layer (region) model in [22]. In this example, which is popular in the NMR and SPT community, one could see that

𝑝(𝐷, 𝜏) changes from a bi-exponential form for small 𝜏 to a mono-exponential form for 𝜏 → ∞. In the intermediate 𝜏-regime fitting multi-exponential behavior is bound to be inaccurate. This example is a simplification of the heterogeneous model introduced in the first experimental part of this contribution, where also a non-trivial 𝜏-dependence of 𝑝(𝐷, 𝜏) is present. The latter provides the information necessary for distinguishing the models leading to the same behavior in the mean square displacement. As an example we mention the form for the van Hove self-diffusion function proposed for heterogeneous systems in [63], which consists of a weighted superposition of Gaussian propagators. In terms of our distribution of diffusivities, Eq. (6), this would result in a superposition of infinitely many, basically exponential contributions of the form given in Eq. (9). With such a superposition almost arbitrarily decaying distributions can be described. Note, however, that the resulting distribution of diffusivities would be again independent of the time lag, 𝑝(𝐷, 𝜏) = 𝑝(𝐷), which neglects that the different diffusive properties typically dominate also on different time scales. An example for the latter for the case of anomalous diffusion is provided below.

3.2.4 Anomalous diffusion processes Diffusion processes in nature often deviate from the mathematical laws of normal diffusion and behave anomalously, a fact which is often reflected in an asymptotically non-linear increase of the meansquared displacement 〈𝒓2 (𝜏)〉⁡~⁡𝐷𝛼 𝜏 𝛼 . In the literature, many theoretical models of anomalous diffusion which can lead to the same diffusion exponent 𝛼 are known [64]. Therefore, the mean-squared displacement is not suitable to discriminate between these models. In this context, an analysis by the distribution of diffusivities provides a significant advantage. To incorporate the non-linear increase of the MSD, the definition of the diffusivities needs to be modified, 𝛽 𝐷𝑡 (𝜏) =

[𝒙(𝑡 + 𝜏) − 𝒙(𝑡)]2 𝜏𝛽

(16)

and we now call them generalized diffusivities. The distribution of generalized diffusivities is defined as 𝛽

𝑝𝛽 (𝐷, 𝜏) = ⟨𝛿 [𝐷 − 𝐷𝑡 (𝜏)]⟩ ,

(17)

where 〈… 〉 again denotes an ensemble average or a time average. If the scaling exponent 𝛽 is chosen to be equal to the diffusion exponent 𝛼 , the mean value of the distribution of generalized diffusivities asymptotically corresponds to the generalized diffusion coefficient 𝐷𝛼 [65].


21

Anomalous diffusion processes often show weak ergodicity breaking, which means that the ensemble-averaged MSD and the time-averaged MSD do not coincide and the latter becomes a random variable, which varies from one trajectory to another, despite the fact that the corresponding state or phase space of the considered system is not divided into mutually inaccessible regions. The reason for the non-ergodic behavior is a diverging characteristic time scale of the process which means that the measurement time can never be long enough to reach this diverging time scale of the system [65]. The probably most investigated example for such a weakly non-ergodic, anomalous diffusion process is the subdiffusive continuous time random walk (CTRW) [18, 66]. For this CTRW, the distributions of generalized diffusivities which are obtained from ensemble and time averages, respectively, strongly differ. However, analytical formulas for the distributions and their explicit 𝜏-dependence are known [65] and can be used to identify this process in experiments. The distribution of diffusivities is a new, promising tool for analyzing data from heterogeneous, anisotropic, and anomalous diffusion processes. It characterizes the diffusivity fluctuations along a trajectory or in an ensemble. For instance, it provides simple means for identifying and characterizing anisotropies in the system. An explicit 𝜏-dependence indicates the presence of a complex underlying process, and its analysis characterizes the latter. Its determination is easily accomplished from experimental data.

4

Conclusions

We have investigated experimentally and theoretically the influence of spatially heterogeneous environments and of anisotropies on the diffusive behavior of particles, as well as the question of ergodicity and weak ergodicity breaking, respectively. These questions were addressed via single particle tracking and NMR experiments, which measure time and ensemble averages, respectively. In this way the question of (weak) ergodicity breaking could be addressed. As a tool for the evaluation of these experimental data the concept of the distribution of diffusivities was applied and its properties were elucidated analytically for various model scenarios relevant in this context.

Acknowledgements The research reported here has been supported by the Deutsche Forschungsgemeinschaft within project P4 “Driven diffusion in nanoscaled materials” of the Saxon Research Group FOR 877 “From Local Constraints to Macroscopic Transport”.

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